{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

pmt1_solution

# pmt1_solution - IEOR 160 Homework#5 Solution 1 A concave...

This preview shows pages 1–2. Sign up to view the full content.

IEOR 160, Homework #5 Solution 1. A concave function cannot have a minimizer over an equality constrained feasible region. False. Consider function f ( x ) = ln x , over the feasible region with equality constraint x = 1. Then x = 1 is a minimizer. If the objective function of an optimization problem is convex and the feasible region is convex, then it is a minimization problem. False. Whether a problem is a maximization problem or minimization problem is nothing related to the property of objective function and feasible region. If f is continuously differentiable function, then all its local maxima is among its stationary points. True. If x is a local maximum of a concave function, then there exists a direction vector d for which the directional derivative at x is negative. False. f ( x ) = 1 is a concave function. And point x = 1 is a local maximum. But for any direction vector d , the directional derivative is 0. For a KKT point, if Lagrangian multiplier of a constraint is zero, then the constraint is inactive at this point. False. For a KKT point, Lagrangian multiplier is zero, the constraint can be either binding or not. For example, max x 2 , constraint is x 0. Then the KKT condition is 2 x + λ = 0, λx = 0, λ 0, x 0. Clearly, ( x, λ ) = (0 , 0) is a KKT point. And λ = 0, but the constraint x 0 is still active at this point. 2. Proof. Use contradiction. Assume that the statement is not true. That it, there is a local minima x of function f ( x ) on S and x is not a global minimum.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern